You have my sympathy, Mart. I had exactly the same problem when I grew potatoes in my garden. I deliberately chose varieties that were described as firm-fleshed or waxy as I love their texture, but they just disintegrated when I cooked them - the outside would be falling away before the inside had cooked. Steaming was a bit better, but still didn't produce the desired texture.
I think the type of soil might have a lot to do with it, as I don't have this problem now I have an allotment. My home soil is very fine, dry, sandy and generally lacking in body (but a delight to dig, even after heavy rain). The allotment soil is heavy, clayey and difficult to work even in ideal weather conditions, but produces great potatoes.
It would be interesting to know if anyone else has found similar differences in potatoes grown on these different soils, or if it is just coincidence.
If soil type is to blame, the best (easiest!!!)solution is probably to get a new plot
Good luck!